| L(s) = 1 | + (0.642 − 0.766i)2-s + (−1.69 − 0.147i)3-s + (−0.173 − 0.984i)4-s + (1.83 + 1.27i)5-s + (−1.19 + 1.19i)6-s + (1.37 + 2.94i)7-s + (−0.866 − 0.500i)8-s + (−0.118 − 0.0208i)9-s + (2.15 − 0.582i)10-s + (3.04 + 1.76i)11-s + (0.147 + 1.69i)12-s + (−1.37 + 0.243i)13-s + (3.13 + 0.840i)14-s + (−2.91 − 2.43i)15-s + (−0.939 + 0.342i)16-s + (−0.343 + 1.94i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (−0.976 − 0.0853i)3-s + (−0.0868 − 0.492i)4-s + (0.820 + 0.571i)5-s + (−0.489 + 0.489i)6-s + (0.518 + 1.11i)7-s + (−0.306 − 0.176i)8-s + (−0.0394 − 0.00696i)9-s + (0.682 − 0.184i)10-s + (0.919 + 0.530i)11-s + (0.0426 + 0.488i)12-s + (−0.382 + 0.0674i)13-s + (0.838 + 0.224i)14-s + (−0.751 − 0.628i)15-s + (−0.234 + 0.0855i)16-s + (−0.0833 + 0.472i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.46551 + 0.0276399i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46551 + 0.0276399i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 + 0.766i)T \) |
| 5 | \( 1 + (-1.83 - 1.27i)T \) |
| 37 | \( 1 + (2.06 + 5.72i)T \) |
| good | 3 | \( 1 + (1.69 + 0.147i)T + (2.95 + 0.520i)T^{2} \) |
| 7 | \( 1 + (-1.37 - 2.94i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (-3.04 - 1.76i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.37 - 0.243i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (0.343 - 1.94i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-7.25 - 0.634i)T + (18.7 + 3.29i)T^{2} \) |
| 23 | \( 1 + (-2.79 + 1.61i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.05 + 7.65i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (2.70 + 2.70i)T + 31iT^{2} \) |
| 41 | \( 1 + (1.78 - 0.315i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 - 3.81iT - 43T^{2} \) |
| 47 | \( 1 + (2.51 - 9.39i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (3.16 - 6.79i)T + (-34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (0.319 + 0.148i)T + (37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (3.68 - 5.26i)T + (-20.8 - 57.3i)T^{2} \) |
| 67 | \( 1 + (14.3 - 6.69i)T + (43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-12.2 + 10.2i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (3.39 + 3.39i)T + 73iT^{2} \) |
| 79 | \( 1 + (1.02 + 2.19i)T + (-50.7 + 60.5i)T^{2} \) |
| 83 | \( 1 + (4.24 + 6.06i)T + (-28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (-6.07 + 13.0i)T + (-57.2 - 68.1i)T^{2} \) |
| 97 | \( 1 + (1.59 + 2.76i)T + (-48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.58267856559500316839844598426, −10.77695201414556510472347093827, −9.677943961742251092877338012654, −9.040017515985167563853779515392, −7.37688416791745073593837036783, −6.12096959889936080982037788181, −5.69375620889480557597299071100, −4.65339275617410850710832217687, −2.94312432957277477708356592041, −1.68144613731945845500996405024,
1.13255759421956534047368256773, 3.45783084221940335720533065856, 5.01027342840490778267301064762, 5.25830511970516025678126417210, 6.53580233718057559308771391906, 7.28322651666772208972999834723, 8.600030457730472759215460212949, 9.550545102605215050246048437878, 10.64424757884641907195831953951, 11.50299073997229824509548214337
